1. Some Documents Containing Advice

The following are a few (fairly detailed and lengthy) documents that contain advice about specific aspects of your graduate career.

After the Master's degree and before the Ph.D.

This is a Guide for Ph.D. Candidates in
Mathematics that I wrote for graduate students shortly
after earning my Ph.D. It contains advice for graduate students in
mathematics during the period of time after they pass qualifying exams
(or preliminary exams) and before receiving their Ph.D. In it you will find
suggestions on such topics as choosing an advisor, beginning to conduct
research, writing up results, and submitting papers for publication.

Having a Grand Project

I gave a talk at the UH Math Department's Graduate Student Seminar
advocating that every graduate student have a "Grand Project", something
special that you choose to work on to make a personal connection with and
contribution to mathematics. Here are the slides from my talk.

Birds and Frogs, an essay by
Freeman Dyson that was written for his planned Einstein Public Lecture. In
it he divides mathematicians into two types: birds, who "fly high in
the air and survey broad vistas" (i.e., seek abstraction, unification, and
generalization), and frogs, who "see only the flowers that grow nearby"
(i.e., study the details of specific examples).

"You and Your Research", a talk
given by Richard Hamming that centered on the question "Why do so few
scientists make significant contributions and so many are forgotten in the
long run?". Hamming discusses what he has learned in terms of the
properties of individual scientists, their abilities, traits, working
habits, attitudes, and philosophies.

Thoughts on Doing Mathematics

Mathematics: What do grad students in math do all day? (and retyped
in PDF) is a response
to this question on Quora by
Yasha Berchenko-Kogan, a Ph.D. student at MIT.
This article discusses how communicating a technical subject (such as mathematics) to a
layperson requires one to avoid technical terms and jargon. These terms have often
been specifically developed or assigned a particular meaning in the discipline for the
purposes of describing aspects
of the subject in a precise way. Therefore, when an expert restricts their language to
common words, it can often feel cumbersome and imprecise, making communication difficult.
This is reminiscent of the xkcd comic
Up-Goer Five in which the artist, Randall Munroe,
explains the blueprints of a Saturn V rocket using only the one thousand most commonly used words
in the English language. (See a
more detailed description of the Up-Goer Five comic here). This comic has led to the
The Up-Goer Five Challenge (also
in PDF) asking scientists to attempt to
explain their own research using only the one thousand most commonly-used words
in English. If you would like to give it a shot, try the Up-Goer
Five Text Editor. You may also want to check out which words comprise
one thousand most commonly used words in English.

3. Professional Societies

There are several professional societies for mathematicians. The two main professional societies are the AMS and MAA.

The American Mathematical Society (AMS) is dedicated to the
interests of mathematical research and scholarship, and serves the national and international
community through its publications, meetings, advocacy and other programs.

The Mathematical Association of America (MAA) works to
advance the mathematical sciences, especially at the collegiate level, through education, research,
professional development, public policy, and public appreciation of mathematics.

Sometimes the differences between the AMS and MAA are summarized as "The MAA is more concerned with mathematics education,
while the AMS is aimed more at professional mathematicians". While there is some truth in this, it is a large oversimplification. The
AMS and MAA are both interested in mathematics education at all levels as well as research mathematics, but their missions place
different emphasis on various issues within education and research.

Since our math department is an AMS department member, all of our math graduate students
receive free membership to the AMS.
This means you should be receiving the AMS Notices and the AMS Bulletin in the mail. If you are not, check with
the department's Director of Graduate Studies regarding your membership.

As a graduate student, you should consider joining other professional societies as well. (I
highly recommend the MAA for anyone planning on a job in academia and SIAM for all applied students or
for students planning a job in industry.) Membership information
can be found on each society's website, and
you should be aware that the student rates for membership are much cheaper than faculty rates.
As a member of a professional society, you
receive the monthly publications of the society, discounts on books and conference
registrations through that society, mathematics and society news,
information about mathematics opportunities, and access to certain online information.
Your membership also supports the mathematics community and
shows a level of professionalism that future employers like to see.

In addition to the AMS and MAA, some other popular mathematics societies with more specialized roles are the following:

The Association for Women in Mathematics (AWM) encourages
women and girls to study and have active careers in the mathematical sciences, and promotes equal
opportunity and the equal treatment of women and girls in the mathematical sciences.

The Society of Actuaries (SOA) is an educational, research, and professional
organization dedicated to serving the public and Society members. The SOA's vision
is for actuaries to be the leading professionals in the measurement and management of risk.

The American Academy of Actuaries (AAA)
The American Academy of Actuaries is a professional association whose mission is to serve
the public and the U.S. actuarial profession. Academy members include consultants,
corporate executives and staff, regulators, government officials, academicians,
and retired actuaries. Their areas of practice cover pensions, life insurance, casualty
insurance, health insurance, financial reporting, risk management, and more.

Even if you are not a member of a particular society, their website can often be a source of very useful
information for you. In particular, the AMS and MAA both have a list of resources for students
on their websites, and these are accessible to everyone.

You can also sign up for the AMS's Headlines
and Deadlines (sign
up here), which is an email notification sent out periodically to
inform you of recent news, helpful websites,
special programs, useful services, announcements and deadlines for
applications to fellowships and grants,
conference announcements, and more.

Student Chapters

Our department has an AMS Graduate Student Chapter (aimed at serving all graduate students), and a
SIAM Graduate Student Chapter (aimed at serving graduate students in applied math). All
graduate students should be involved in these two student chapters and regularly attend the
events that they offer.

4. Online Resources for Research

Books and Papers

The arXiv preprint server. Almost all
mathematicians post their preprints on the arXiv, as well as updates to the preprints. You can
search, browse the
new and
recent submissions, or
subscribe to a mailing list to
be updated on recent submissions. You can also do everything
by category. Once you have
decided on an advisor and research area, you should
subscribe to the arXiv mailing list in the relevant
categories. You will receive an email about once per week with all new submissions and
updated replacement submissions in your chosen categories. This is a great help for
keeping updated on progress and advancements in your field. (Note: The arXiv itself is located at
http://arxiv.org, but a front end that is
sometimes easier to navigate is maintained by UC Davis at
http://front.math.ucdavis.edu.)

MathSciNet. You can look up published papers and read reviews on MathSciNet, as well as
obtain information for bibliographies of papers you may be writing. If our library has
an electronic subscription to a journal, you can also access electronic copies of
papers through MathSciNet. (Note that MathSciNet requires a subscription, so to use it you must
login through the library using your CougarNet ID.)

ILLiad. If you need a paper or other
source that you cannot get online, from the arXiv, or through MathSciNet, you can get almost
anything you need through ILLiad, UH's InterLibrary Loan system, which is known as Illiad.
If you request a paper or book chapter, it will usually be delivered to you in PDF
format within 24 hours. If you need a book, it will take a bit more time
because it must be delivered physically.

Answering Questions

MathOverflow is a question-and-answer
web site for research-level math questions or other
questions of interest to professional mathematicians.

Wikipedia actually has a
number of very well written articles on advanced mathematical topics. These articles can often
be used to answer basic questions. However, keep in ming that as with any
encyclopedia, Wikipedia is a starting point ---not an ending point--- for academic research.
Also, you may ask: "Should one cite Wikipedia in research papers?". Wikipedia says "no", and
here is a Wikipedia
article discussing the issue.

Advice for Reading Papers

Here is some advice for students reading papers for the first time:

Begin with the abstract to see if you're interested.

Read the introduction and conclusions with extra focus on the main ideas and results.

Skim the middle sections to get a feel for the flow, pay special attention to main
theorems.

Read select parts in more detail. Feel free to skip around depending on your level
of interest and how much detailed info you need.

When an especially deep understanding is required, it may be necessary to go back
and read the entire article from start to finish.

5. Attending and Giving Talks

Attending Talks, Colloquia, and Seminars

Colloquium. All graduate students should regularly
attend the math department colloquia.

Research
Seminars in the Mathematic Department. These talks are usually fairly specialized.
In your first and second years of graduate school, you can sample a few to help you decide
which area you want to go into. After you choose a research area and advisor, you should
regularly attend any seminar series relevant to your area.

The Graduate Student Seminar and the AMS and SIAM chapters have talks aimed at graduate students
and should be accessible, relevant, and interesting to all graduate students. You should do your
best to attend all of these.

Unfortunately, many colloquia and seminar talks are often bad; usually because they are too technical
and difficult for non-experts to follow. Despite this, it is still beneficial for you and necessary for your
professional development that you attend them. (Remind yourself that one useful talk is often
worth attending ten bad talks.)

Try to get something out of every talk. Gain exposure to new ideas. Learn what topics are
at the forefronts of research right now. Learn the concepts and words that come up in
current research. Learn (perhaps by counterexample) what the
elements of a good talk are. In addition, strive to understand at least some small part of
every talk you attend. Here is some advice for doing so:

Mathematical Writing by Donald
E. Knuth, Tracy Larrabee, and Paul M. Roberts
are the notes/text from a course on writing given by the authors at Stanford University
in Fall 1987. In addition, here is a list of errata.

Teaching Experiences at UH

During your time in graduate school, you should make a conscious
effort to get as much experience and training in
teaching as you can. This will not only help you in your future career, but it will
help you get a job as you enter the highly competitive academic job market.

If you intend to have a career in academia, you should seek out additional teaching opportunities
beyond your TA duties. Here are a few options.

CHAMP is an outreach program in the math
department that can give you teaching experience either as a facilitator or an instructor.
If you are interested in participating in CHAMP,
please contact me.

Summer Teaching Opportunities

If you want to use the summer to gain teaching experience, there are many K-12 mathematics
summer programs that seek (and pay) graduate students and faculty to assist with teaching.
Here are a few:

Some Advice as You Develop Your Teaching Skills

In addition to lecturing, experiment with
nontraditional teaching methods; e.g., discovery-based learning,
inquiry-based learning, modified Moore methods, group work, interactive techniques such as
the use of "clickers", assigning
projects or writing activities, or using technology in the classroom.

Form a Teaching Group or Teaching Seminar; i.e., find a group of like-minded students, postdocs, or faculty
interested in teaching who would be willing to meet regularly to discuss
teaching methods and experiences in their classrooms that semester.

8. Summer Programs for Graduate Students

Park City Math
Institute (PCMI) is a 3-week mathematics program held at Park City, Utah.
The mathematical topic changes from year to year. Participating groups
include mathematics educators, undergraduate students, graduate students, and
mathematics researchers. There are individual courses of study within each of
these groups, as well as a meaningful amount of interaction among the groups.
The rich mathematical experience
combined with interaction among all participants results in greatly
increased understanding and awareness of the issues confronting
mathematics and mathematics education today.

AMS-AAAS
Mass Media Fellowship is a highly competitive program designed to improve public
understanding of science and technology by placing advanced science, mathematics, and engineering students
in newsrooms nationwide. Fellows work with media professionals to improve their communication skills and
increase their understanding of the editorial process by which events and ideas become news.

10. Harsh Realities

"[O]f all the machines that humanity has created, few
seem more precisely calibrated to the destruction of
hope than the academic job market." ---Dr. Patrick Iber (At the time he wrote these words, Dr.
Iber had a Ph.D. from the University of Chicago, a book
contract with Harvard University Press, and a visiting
lectureship at UC Berkeley.)

The drop-out rate in graduate programs throughout the U.S. has become increasingly high,
and the job market for Ph.D.s is not particularly good, especially for academic jobs.

Furthermore, in academic jobs, the path to tenure has become longer and more difficult
over the past few decades, with greater emphasis placed on obtaining grants and
external funding. In addition, the hiring of more and more adjunct faculty and
lecturers has led to a decrease in the number of faculty positions and more
exploitation of academics with Ph.D.s.

On top of all this, the recent economic problems combined with the sequestration imposed by
the U.S. government has made a bad situation worse, and created an environment of scarcity for
both academic and non-academic jobs.

It is important to be aware of these issues and prepare for them The
information here is meant to help you, not to discourage you. It is intended to help you make
an accurate assessment of the current state of graduate programs and job markets for Ph.D.s, so
that you can make informed decisions and plan for the future. It is important for all
graduate students to start
preparing for jobs and gaining skills to make themselves marketable in the
early stages of their graduate career, long before earning their degree and applying for jobs.

Thoughts on Graduate School

100 Reasons NOT to Go to Graduate School.
Although stated somewhat negatively, the posts on this site describe many difficult and problematic
aspects of academia in a very accurate and even-handed way. Although its focus is on the
humanities and social sciences, almost everything the blog discusses is relevant to all
academic disciplines, including mathematics.

Data on the Profession

If you look at the "Mathematics" row in the following chart of
Data on Doctorates Awarded, you can see that the number of Ph.D.s
awarded in the U.S. (to both U.S. citizens and non-U.S. citizens) was 1,050 in 2000 and increased steadily
to 1,554 in 2009. In addition, although not shown in that report, there were 1,798 Ph.D.s awarded in 2012.
Note that this is
an increase of approximately 80% in the number of Ph.D.s awarded from 2000 to 2012. Considering how the
number of mathematics Ph.D.s awarded is increasing, but the number of jobs
(particularly faculty positions) for mathematics Ph.Ds is decreasing, you can get a
sense of the difficulties in the mathematics Ph.D. job market.

Total Ph.D.s Awarded in 2012: 1,798
Of those, the number awarded to . . .

U.S. Citizens: 863 (621 men and 242 women)
U.S. Citizens who are Black: 28 (16 men and 12 women)
U.S. Citizens who are Hispanic: 33 (22 men and 11 women)
U.S. Citizens who are Pacific Islanders: 5 (3 men and 2 women)

If you are a graduate student who is a member of an underrepresented group (e.g., a woman or minority)
consider taking advantage of opportunities and organizations designed to encourage and support your
professional development. The AMS and MAA have a number of resources. Also, consider being involved in the
Math Alliance or the AWM.
If you are interested in talking more about resources for members of underrepresented groups, feel free to
contact me.

Interdisciplinary Work

Interdisciplinary Penalty (also in PDF) is an article published
by Inside Higher Ed based on this study, which states that Ph.D.s who write
interdisciplinary theses tend to make less money. Students in Math
Biology or other interdisciplinary subjects should take note of some of the points.

This Ph.D Comics news story discusses how
funding for basic science in the U.S. is currently the worst it has ever been, and includes
interviews with scientists discussing how this has affected their research and academic career plans.

Postdoctoral Positions and Tenure

To get a tenure-track job at a research university, you will need to do at least one postdoc. The
following comic strips from Ph.D.Comics give you a sense of the
impermanence of these positions, and the uncertainty of future employment, as you try to get your next position.

The Perils of Tenure

"The Tenure Chase Papers" is a memoir by Dana Mackenzie about his candidacy for tenure at Kenyon College,
which ultimately ended with him being denied tenure and released from his position.

From author: "The Tenure Chase Papers" is a tale that is full of unexpected twists and turns and
good lessons for young professors on the tenure track. It draws back the veil of
secrecy surrounding the most critical career hurdle for anybody in academia.
It sometimes seems taboo to speak of tenure, but when things are left unspoken,
lessons are left unlearned. . . . I still think that there are very serious questions
about the wisdom of the tenure system.
Does this all-or-nothing hurdle make sense in the modern world? Suppose we concede for a
moment that universities should grant tenure only to faculty who are truly outstanding in
every way. Why should the penalty for being not-quite-outstanding be that you are essentially
fired? I don't know any other profession that conducts its business in such a self-defeating way.

The website www.phds.org said "[The Tenure Chase Papers] is required reading
for all academics and would-be academics". The "Tenure Chase Papers" was published in the book
Starting Our Careers (American Mathematical Society, 1999), pp. 79-100.
You can also read it on
Dana Mackenzie's Website or here: